What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx 1{,}000{,}000$ elements.
I'll start the ball rolling with the five platonic solids. (Unfortunately not large.)
On
As is well known, every finite (natural) number can be associated with a finite set of that cardinality. So in particular, the cardinality of a famous or special finite set must be a famous or special number. Here's a list of all the special numbers less than or equal to 9999 and contains quite a few items between $2^{10}$ and $2^{20}$.
Oh, what, you actually want the sets, and not just their cardinality, because there is more than one way of realising a set of a given cardinality? (Me grumbles something about bijective maps and isomorphisms of sets.) Fine:
As beauty is in the eye of the beholder, I'm sure there are mathematicians out there who think each of the above numbers ought to be better known.
The sporadic groups? In particular they are finite sets... quite a few are too big to fit into your range, but the smallest (Mathieu groups) would do the trick.
http://en.wikipedia.org/wiki/Sporadic_groups